Applied Mathematics, 2013, 4, 1340-1346
http://dx.doi.org/10.4236/am.2013.49181 Published Online September 2013 (http://www.scirp.org/journal/am)
Randomly Weighted Averages on Order Statistics
Homei Hajir, Hasanzadeh Leila, Mina Ghasemi
Department of Statistics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
Email: homei@tabrizu.ac.ir.
Received July 23, 2012; revised January 4, 2013; accepted January 11, 2013
Copyright © 2013 Homei Hajir et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
We study a well-known problem concerning a random variable uniformly distributed between two independent random
variables. Two different extensions, randomly weighted average on independent random variables and randomly
weighted average on order statistics, have been introduced for this problem. For the second method, two-sided power
random variables have been defined. By using classic method and power technical method, we study some properties
for these random variables.
Keywords: Two-Sided Power; Moment; Weighted Averages; Power Distribution
1. Introduction
Van Asch [1] introduced the notion of a random variable
Z uniformly distributed between two independent ran-
dom variables X1 and X2 which arose in studying the dis-
tribution of products of random 2 × 2 matrices for sto-
chastic search of global maxima. By letting X1 and X2 to
have identical distribution, he derived that: 1) for X1 and
X2 on [1, 1], Z is uniform on [1, 1] if and only if X1
and X2 have an Arcsine distribution; and 2) Z possesses
the same distribution as X1 and X2 if and only if X1 and X2
are degenerated or have a Cauchy distribution. Soltani
and Homei [2] following Johnson and Kotz [3] extended
Van Asch’s results. They put 1,,
n
X
X to be inde-
pendent, and considered
112211, 2
nnn
SRXRXRX RXn
−−
=+ +++≥nn
U
.
where random proportions
()( )
1, 1,,1,
iii
RU Uin
=−= −
()( )
1
1
0
1, ,,
n
ni n
i
RRU
=
=−
are order statistics from a uniform distribution on [0, 1],
and
()
0. These random proportions are uniformly
distributed over the unit simplex. They employed Stielt-
jes transform and that: 1) n possesses the same distri-
bution as 1
0U=
S
,,
n
X
X if and only if 1,,
n
X
X are de-
generated or have a Cauchy distribution; and 2) Van
Asch’s result for Arcsine holds for Z only.
In this paper, we introduce two families of distribu-
tions, suggested by an anonymous referee of the article,
to whom the author expresses his deepest gratitude. We
say that Z1 is a random variable between two independ-
ent random variables with power distribution, if the con-
ditionally distribution of Z1 given at 112
, 2
X
xX x== is
()
()
()
11 2
12
1
12
21
,
1
21
21
12
1max
,
1,
1max
,,
n
Zxx zn
zxx
zx xzx
xx
Fzx xzx
xx
zx

<<

 
=
 
−<<


,.x
(1.1)
The distribution
()
11 2
,Zx x
F
z
2
will be said to follow a
conditionally directed power distribution, When n is an
integer. For n = 1, the distribution given by (1.1) simpli-
fies to the distribution Z that was introduced before. Also
we used Stieltjes methods, for more on the Stieltjes
transform, see Zayed [4].
For n = 2, we call Z1 directed triangular random vari-
able. For further generalizing Van Asch results, we in-
troduce a seemingly more natural conditionally power
distribution. We call Z2 two-sided power (TSP) random
variable if the conditionally distribution of Z
2 given at
112
,
X
xX x== is
()
21 2
2
1
1
,
21
1
1,
.
0.
n
Zxx z
zy
zy
2
F
yzy
yy
zy

=<


< (1.2)
C
opyright © 2013 SciRes. AM
H. HAJIR ET AL. 1341
The distribution 212
,
Z
xx will be said to follow a con-
ditionally undirected power distribution, when y1 = min
F
()
11
min,2
y
xx=,
()
,
212
max
xx= and n is an integer.
Again for n = 1, the distribution given by (1.1) simpli-
fies to the distribution Z that was introduced by Van Asch.
The main aim of this article is providing a generalization
of notion to the results of Van Asch for some other values
of n (other than n = 1). This article is organized as follows.
We introduce preliminaries and previous works in Section
2. In Section 3, we give some Characterizations for dis-
tribution of Z1 given in (1.1), when n = 2. In Section 4, we
find distribution of Z2 given in (1.2) by direct and power
method, and give some examples and Characterizations of
such distributions by use of Soltani and Homei’ results [5].
2. Preliminaries and Previous Works
In this section, we first review some results of Van Asch
[1] and then modify them a little Bit to fit in our frame-
work, to be introduced in the forthcoming sections. Us-
ing the Heaviside function ,
we conclude that for any given distinct values X1 and X2
the conditional distribution
()
()
0, 0,1, 0Uxxx x=<=≥
()
11 2
,Zx x
F
z in (1.1) is
() ()
()
11 2
1
1
,
21
1
2
121
.
n

Zxx
i
n
i
zx
Fz Uzx
xx
nzx Uzx
ixx
=
=−




−−




(2.1)
Lemma 2.1. For distinct real’s x1, x2, z and integer n,
we have
()( )
()
() ()()
()()
1
1
1212 21
2
12
1
1d1
1!
d
1.
nn
n
n
zxx xnzxx x
x
xzxz

+⋅


−− −−−

=−−
1
Proof. It follows from the Leibniz formula.
Let , where is an interval, and
is the set of all real functions f that are -
()
hDI
α
)
II
(
D
α
α
Times differentiable on I. If
() ()
1
gz zx
=. for some
constants c and
{
}
1, ,kn
. Then
()()()() ()
{}
() ()
{}
1
1
dd
dd
d.
d
kk
kk
k
k
PMgzhz kgzhzgzP
zz
hzgz
z
=− +−
=
We use the Leibnitz formula for the th deriva-
tive of a product, namely
(
1k
)
() ()
{}
() ()
1
1
1
0
d
d
1dd.
dd
k
k
iki
k
iki
i
hzgz
z
k
Nhz
izz
=


== 



Let
() ()
{}
() ()
{}
1
1
ddd .
d
dd
kk
kk
hzgzhzgzM N
z
zz
==+
where
() ()
11
1
11
0
1dd,
dd
iki
k
iki
i
k
M
hz gz
izz
+−−
+−−
=
 

=
 



() ()
1
0
1dd
dd
iki
k
iki
i
k
Nhz
izz
=
gz
 

=
 



.
Since
() ()
1
d.
d
r
r
!r
gz
zzx
+
=r
It follows that
() ()()
1
1
dd
dd
rr
rr
g
zrgz gz
zz
=

. Consequently,
()() ()
{}
1
1
d
d
k
k
N kgzhzgzP
z
=− where after some al-
gebraic work
() ()
dk
PMgz hz=− dk
z.
Therefore,
() () () ()
{}
1
1
dd .
dd
kk
kk
MNgzhzkhzgz
zz
+= +
This completes the proof.
Another tool for proving our main theorem is the fol-
lowing formula taken from the Schwartz Distribution
theory, namely,
()
[]
() () ()
1d
d!d
nn
n
n
x
x
nx
ϕϕ
∞∞
−∞ −∞
Λ= Λ

xx (2.2)
where is a distribution Function and is the n-th
distributional derivative of .
Λ
[]
n
Λ
Λ
The conditional distribution
()
11 2
,Zxx
F
z given by (1.1)
leads us to a linear functional on complex Valued func-
tions f: , defined on the set of real numbers :
()
()
()
()( )()
11 2
,
1
2
1
21 21
1d .
d
!
Zxx
ni
n
nin
i
Ff
fx
i
f
x
z
xx nixx
=
=−
−−−
It easily follows that
() () ()
11 211 2112
,, .
ZxxZxx Zxx,
F
af bgaFfbFg+= + (2.3)
For any complex-valued functions f, g and complex
constants a, b. We note that
() ()
11 211 2
,,
,
z
Zxx Zxx
F
zF f=
Whenever
()()(
n
z
)
f
xzxUzx=− − and
()
gz
()
()
()( )()
11 2
,
1
2
1
21 21
1d .
d
!
Zxx
ni
n
zz
nini
i
fx
Ff
f
x
z
xx nixx
=
=−
−−−
Copyright © 2013 SciRes. AM
H. HAJIR ET AL.
1342
Also we note that
()
() ()
1d.
!d
nn
z
n
Uz xfx
n
x
−= Thus
()()()
()
()
1
11 2
2
1
2
,
1
d
d,
i
Z
Xi
Zxx i
PZzUz xFx
F
zFx
=
≤= −
=
can be viewed as:
() () ()
()
()
1
11 2
2
2
,
1
1dd
!d
d.
i
nn
zZ
n
z
Xi
Zxx i
f
xFx
nx
F
fFx
=
=
(2.4)
Therefore by using (2.3) along with (2.4) and a standard
argument in the integration theory, we obtain that
() () ()
()
()
1
112
2
2
,
1
1dd
!d
d.
i
nn
Z
n
Xi
Zxx i
f
xFx
nx
F
fFx
=
=
(2.5)
For any infinitely differentiable functions f for which
the corresponding integrals are finite. Now (2.5) together
with (2.2) lead us to
()
()
() ()
()
111 2
2
2
,
1
d
i
nd.
Z
Xi
Zxx i
f
xFxFfF x
=
=

(2.6)
For the above mentioned functions f, where 1
()
n
Z
F
is the
(n)-th distributional derivative of the distribution of Z1.
Let us denote the Stieltjes transform of a distribution H
by
()
()
()
1
,d.
SHzH x
zx
=
For every z in the set of complex numbers which
does not belong to the support of H, i.e., .
()
suppHc
z
The following lemma indicates how the Stieljes trans-
form of Z1 and X1, X2 are related.
Lemma 2.2. Let Z1 be a random variables that satisfies
(1.1). Suppose that the random variables X1 and X2 are
independent and continuous with distribution functions
1
X
F
and 2
X
F
respectively. Then
()
()()
()
()
()
11
1
1,,
suppH .
nn
ZXX
c
SFz SFzSFz
n
z
=−
2
,,
Proof. It follows from (2.6) that
()
()
()
()
111 2
2
2
,
1
,
i
nd
Z
zX
Zxx i
SFzFgF x
=
=
i
.
And
()
()
()
111 2
2
2
,
1
1d ,d
!d i
n
Z
zX
Zx x
ni
SF zFgFx
nz =
=
i
for
()
1
z
gx zx
=. Now, it follows that
()
()()()
11 2
,
1
12
21 21
1
1d1
.
d
!
z
Zxx
ni
n
ni
i
Fg
zx
zzx
xx nixx
=
=−
−−−
ni
And by using Lemma 2.1, we have
() ()
()()
11 2
,
12
1.
n
z
Zxx n
Fg
zxzx
=−−
Therefore,
()
()
()()
()
1
2
2
1
12
1
1d ,d
!di
n
n
ZX
nn
i
SF zFx
nzzx zx=
=−−
,
i
and
()
()()
()
()
()
11
1
1,,
suppH .
nn
ZXX
c
SFz SFzSFz
n
z
=−
2
,,
)
2
,
(2.7)
This finishes the proof.
Note that Van Asch’s lemma is the case of n = 1:
()()(
11
,,
ZXX
SF zSFzSFz
−= .
We also note that the Stieltjes transform of Cauchy
distribution, i.e.,
()
1
,SFz zc
=+ satisfies (2.7).
3. Characterizations
Now, we apply Lemma 2.2 for some characterizations,
when X1 and X2 are not identically distributed.
Theorem 3.1. Let X1 and X2 be independent random
variables and Z be a randomly weighted average given in
(1.1).
For n = 2 we have,
a) if X1 has uniform distribution on [1, 1], then Z1 has
semicircle distribution on [1, 1] if and only if X2 has
Arcsin distribution on [1, 1];
b) if X1 has uniform distribution on [1, 1], then Z1 has
power semicircle distribution on [1, 1] if and only if X2
has power semicircle distribution i.e.,
()
()
2
31 , 11
4
z
f
zz
=−≤≤
.
c) if X1 has Beta (1,1) distribution on [0, 1], then Z1 has
Beta 33
,
22

distribution if and only if X2 has Beta
Copyright © 2013 SciRes. AM
H. HAJIR ET AL. 1343
11
,
22

distribution;
d) if X1 has uniform distribution on [0, 1], then Z1 has
Beta (2, 2) distribution if and only if X2 has Beta (2, 2)
distribution.
Proof. 1) For the “if” part we note that the random
variable X1 has uniform distribution and X2 has Arcsin
distribution on [1, 1]; then
()
()
1
1
,ln1ln
2
X
SF zzz=+−−
1.
And
()
22
1
,.
1
X
SF zz
=
From Lemma 2.2 and substituting the corresponding
Stieltjes transforms of distributions, we get
()
()
13
22
2
,.
1
Z
SFz
z
′′ =
The solution
()
()
1
2
,2 1.
Z
SF zzz=−−
Which is the Stieltjes transform of the semicircle dis-
tribution on [1, 1].
For the “only if” part we assume that the random
variable Z1 has semicircle distribution. Then it follows
from Lemma 2.2 that
()
()
223
22
11
,11
X
SF zzz
=
.
The proof is completed.
2) By an argument similar to that given in 1) and
solving the following differential equations,
()
()
()
()
()
1
2
,
26lnln16
1
Z
SFz
zz zzz
zz
′′
=−−−−
3.+
(for the “if” part), and
()
()
()
2
2
S,6 lnln16
X3
F
zzzzzz=− −−+−.
(for the “only if” part).
The proof can be completed.
3) By Lemma (2.2), we have
()
()()
11
,,
211
Z
SFz zz zz
′′
−=
−−
(for the“if” part), and
()
()
()()
2
11
,
111
X
SF z
zz zz zz
−=
−−
,
(for the “only if ” part).
The proof can be completed by solving the above dif-
ferential equations.
4) By Lemma (2.2), we have
()
()
()
()
()
1
2
,
26lnln16
1
Z
SFz
zz zzz
zz
′′
=−−−−
3+
(for the “if” part), and
()
()
()
2
2
S,6 lnln16
X3
F
zzzzzz=− −−+−
(for the “only if ” part).
Solving the differential equations, can complete the
proof.
4. TSP Random Variables
In Section 3, we used a powerful method, based on the
use of Stieltjes transforms, to obtain the distribution of z1
given in (1.1). It seems that one can not use that method
to find distribution of z2 given in (1.2). So we employ a
direct method to find the distribution of z2. Let us follow
Lemma 4.1 to find a simple method to get the distribu-
tion of z2 following [2] and the work of them leads us to
the following lemma.
Lemma 4.1. Suppose W has a power distribution with
parameter n, n 1, n is an integer, and let
212
=1,,
n
()
11
min ,yXX=,
()
,yX
2
max X where
X
X
random variables are. Let independent
()
121
X
YWYY=+−.
Then
1) X is a TSP random variable.
2) X can be equivalently defined by
()
12 12
11
22
X
XX WXX

=++−−

 .
Proof. 1)
()( )
()
()
()
12 121 112
,
1
121
21
,
.
Xx x
n
2
F
zPYWYYzXxXx
zy
Py Wyyzyy
=+ −≤==

=+ −≤=


Proof. 2)
() ()
11221212
, min,,max,,XXx XxUxxxx==
()
() ()
[]
12
12 12
min ,0,1 ,
max ,min ,
Xxx
WU
xx xx
=
and also
()
12 12
12
min ,,
2
x
xxx
xx +−−
=
()
12 12
12
max ,.
2
xxx
xx ++−
=
Copyright © 2013 SciRes. AM
H. HAJIR ET AL.
1344
then
12
12
12
22
,
x
x
xx
X
Wxx
+
−−
=
so
()
12 12
11
.
22
X
xx Wxx

=++−−


4.1. Moments of TSP Random Variables
The following theorem provides equivalent conditions
For .
2
Theorem 4.1.1. Suppose that z2 is a TSP random vari-
ablesatisfying (1.2). If X1 and X2 are random variables
and
k
XEz
μ
=
k
i
Ex =∞, for all integers k then 1, 2i=
()
1)
()
()
()
()
2
0
1
11
k
i
Ez nEy y
nk ki
=
=Γ+
+ Γ−+
12
i
ki
k
kkin−Γ+ Γ+.
2)
()
21
11
22
ki iiki
kk
EzEWEX XX X
i

 
=−+

 
 
 212
+
.
3)
()
()
21
i
kki
kn
EzE yyy
ini


=−


+

 21
.
Proof. 1) By using Lemma 2.1, we obtain that
()
()
()
()
()
()
()
()
()
()
()
21
0
21
0
12
0
1
1
1.
11
ki
kiki i
i
ki
kiki i
i
kiki
i
kEWWEY Y
i
kEWWEY Y
i
kkin
n
nk ki
−−
=
−−
=
=

=−



=−


Γ+ Γ−+
=Γ++ Γ−+
E
yy
Proof. 2) This can be easily proved by Lemma 4.1 2).
Proof. 3)
()
()
()
()
()( )
()
21 21
121
0
121
0
k
k
ki
iki
i
kii
ki
i
EzYW YY
k
EWyyy
i
kEWE yyy
i
=
=
=+ −

=−



=−


()
()
21
i
kki
kn
EzE yyy
ini


=


+

 21
.
Let us consider expectation and variance of z2. First, we
suppose that 11
EY
μ
=, 22
EY
μ
=
12
σ
, ,
, and . Then
2
11
VarY
σ
=2
Y=Var
2
2
σ
()
12
,YY =Cov
12
21
n
EZ n
μμ
+
=+,
and also, if then
12
0EXEX==
() ()
21 2
1
n
E ZEYEYEY
n
=+ −
+1
.
By 1212
X
XYY+=+. We have
() ()()
21 1
1
2
11
nn
EZ EYEYEY
nn
=+−=
++
1
. (4.2)
It can easily follow from (4.2) that the Arcsin result of
Van Asch [1] is only true for n = 1, about the variance, we
have
( )()()
()
()( )
2
22
22
1221 12
2
Var
121.
12
Z
nnnnn
nn
μμσσ σ
−++++ +
=++
Following the computation of expectation and vari-
ance, we evaluate them for some well-known distribu-
tions. If X1 and X2 have standard normal distributions,
then from Theorem 3.1.1 2) and the fact that 12
X
X
and 12
X
X+ are independent, it follows that their first,
second and third order moments are equal, respectively,
to
2
11
1
π
n
EZ n

=
+

,
()( )
2
2
2
2
12
nn
EZ nn

++
=

++

,
and
()()()
32
3
2
151213 30
312
2π
nnn
EZ nnn

++−
=

+++

.
Also, in case X1 and X2 have uniform distributions,
Theorem 4.1.1 2) implies that,
()
()
()
()
()()
2
0
12
112
k
k
i
kkin
Eznnkkiki
=
Γ+Γ−+
=Γ++Γ−++ +
1
.
()
2
21
31
n
EZ n
+
=+,
and
()()
32
22
136
Var 18 11
nnn
Znn
+++
=++
2
.
Theorem 4.1.2. Suppose that z2 is a TSP random
variable satisfying (4.1), then
1) z2 is location invariant;
2) If X1 and X2 have symmetric distribution around
μ
,
then z2 has symmetric distribution around
μ
, only when
n = 1.
Proof.
1) Is immediate.
2) We can assume without loss of generality that 0
μ
=
If Z2 has a symmetric distribution around zero, then
Copyright © 2013 SciRes. AM
H. HAJIR ET AL. 1345
() (
121121
d
YWYY YWYY+−=−+−


)
.
We note that
() ()
()
1211 21
d
YWYYYWY Y

+−=−+−−−

.
Since
()(
1212
min ,max,
)
X
XX−=−X
11
d
,
X
X=− ,
And 22
d
X
X=− , we have
() (
12121
d
YWYYYWYY+−=+−
)
2
. (4.3)
By equating the conditional distributions given at
11
X
x= and 22
X
x= in (3.3), we conclude that n = 1 It
can also easily follow from Theorem (4.1.1) that the
Cauchy result of Van Asch [1] is true only for n = 1.
4.2. Distributions of TSP Random Variables
In this subsection, we investigate computing distributions
by the direct method. We will give two examples of
derivation based on (4.1). This method may be compli-
cated in some cases, but we have chosen some easy to
find examples. We use randomly weighted average on
order statistics to find the distribution of z2. Gauss hyper
geometric function
(
,,;
)
F
abcz which is a well-known
special function that we used in this way.
Example 4.2.1. Let X1, X2 and W be independent ran-
dom variables such that X1 and X2 are uniformly distrib-
uted over [0, 1], and W has a power function distribution
with parameter. We find the value
(
2;
)
Z
f
zn by means
of
()
2z;
Z
Ww
f therefore
()
()
2
20,
21 1.
1
ZWzw
zzw
w
fzwz
w
<<
=
<<
)
(4.4)
By using the distribution of W, the density function
2
(
;
Z
f
zn , can be expressed in terms of the Gauss hyper
geometric function
(
,,;
)
F
abcz which is a well-known
special function. Indeed according to Euler’s formula, the
Gauss hyper geometric function assumes the integral
representation
()
()
() ()()( )
11
1
0
,,;
11
cb a
b
F abcz
ctt tz
bcb
−− −
Γ
=−
ΓΓ−
d,t
where a, b, c are parameters subject to ,
, whenever they are real and z is the variable.
a−∞ <<∞
0cb>>
()
()
()( )
2
1
;
2121 1,,1,
1
Z
nn
fzn
nz zzzFnn
n
=−+−+
where n > 0 and . There are some important func-
tions as a Gauss hyper geometric function.
1n
,z
)
(4.5)
() (
log11,1; 2;zzFZ+= −.
elim ,;;.
zbz
Fabb
→∞

=
a

() ()
1, 1; 1;.
a
zFaz
−=
When n = 1 similar calculations lead to the following
distribution
()( )()
2log12log, 01
Z
( )
21
f
zzzzz=−−−<<z.
When n is an integer, we obtain the following distribu-
tion.
()
()
() ()
()
()
2
1
1
0
,2 1
1
11
211 1
01.
n
Z
i
ni
i
n
fznzz
n
n
nz zz
ii
z
∗−
=
=−

−− −−


<<
,
The probability density function 2
Z
()
f
z was introduced
by Johnson and Kotz [3], for the first time, under the title
“uniformly randomly modified tine”. So 2
(
;
Z
)
f
zn can
be seen as an extension of the above mentioned distribu-
tion. We note that, from (4.1) and a simple Monte Carlo
procedure using only simulated uniform variables, one
can to simulate the distribution (4.5).
Theorem 4.2.2. Let z2 be a undirected triangular ran-
dom variable that satisfies (1.2). Suppose that the random
variables X1 and X2 are independent and continuous with
the distribution Functions 1
X
F
and 2
X
F
, respectively.
Then
()
()()(
12 12
1,,,2,
ZXX XX
z SFzSFzSFFz
′ ′
−= +
)
,,
2SF
′′′
where
()
()()()
()
12
2
2
2
1
1221
,,
1d.
i
XX
Xi
i
SF Fz
F
x
zx zxxx=
=−− −
Proof. By using an argument similar to that given in
section 3, we can conclude that
()
()
() ()
()
221 2
2
2
2
,
1
dd
i.
Z
Xi
Zxx i
f
xF xFfFx
=
=

So,
()
()
()
221 2
2
2
,
1
1,d
2i
Zz
Zxx i
SFzFgF x
=
′′′
−=
,
Xi
For
() ()
2
1.
z
gx zx
= From
Copyright © 2013 SciRes. AM
H. HAJIR ET AL.
Copyright © 2013 SciRes. AM
1346
()()
()
()
()
21 2
22
12
,22
21 12
11
.
z
Zxx
zx zx
Fg
xx xx
−−
=+
−−
() () () (
1
1
0
)
0>
1
,1d, ,
,
xb
a
x
Iabtttab
Bab
=−
.
5. Conclusion
And by using partial fractional rule, we have
We have described how directed methods could be used
for obtaining the distributions, Characterizations and
properties of the random mixture of variables defined in
(1.1). The TSP random variable when X1 and X2 have
uniform distributions, led us to a new family of distribu-
tion which can be regarded as some generalization of
“uniformly randomly modified tine”. The proposed
model in the direct method can easily lead to distribution
generalizations, though this is not possible for the Stielt-
jes method, but here the characteristics can be easily
computed.
()
()()
()
()()
21 2
,
22 2
12
1221
121
.
z
Zxx
Fg
zx zx
zxzxx x
=+
−−
−− −
Therefore,
()
()()
()
()()()
2
2
22
12
2
2
1
12
21
11
,
2
21 d.
i
Z
Xi
i
SFz zxzx
F
x
zx zx
xx =
′′′
−=
−−
+−−
6. Acknowledgements
And
()
()()(
12 12
1,,,2,
2ZXX XX
SFz SFzSFzSFFz
′′′′ ′
−=+
)
,.
The author is deeply grateful to the anonymous referee
for reading the original manuscript very carefully and for
making valuable suggestions.
This finishes the proof.
It is worth mentioning that the present method yields
other extensions too; the following is such an example. REFERENCES
Example 4.2.3. Suppose that X1, X2 W are independent
random variables. If X1 and X2 have Uniform distributions
on [0, 1] and W has Beta (2, 2) distribution, then z2 has the
same distribution as W.
[1] W. Van Asch, “A Random Variable Uniformly Distrib-
uted between Two Independent Random Variables,”
Sankhaya, Vol. 49, No. 2, 1987, pp. 207-211.
doi:10.1080/00031305.1990.10475730
If the product moments of order statistics are known,
those of W can be derived from that of z2. By using
Theorem 4.1.1 1). Then the distribution of W is charac-
terized by that of z2.
[2] A. R. Soltani and H. Homei, “Weighted Averages with
Random Proportions That Are Jointly Uniformly Distrib-
uted over the Unit Simplex,” Statistics & Probability
Letters, Vol. 79, No. 9, 2009, pp. 1215-1218.
doi:10.1016/j.spl.2009.01.009
By an argument similar to the one given in example
4.2.1, when W has a Beta distribution with Parameters n
and m, we find the distribution
()
2;,
Z
f
znm as
[3] N. L. Johnson and S. Kotz, “Randomly Weighted Aver-
ages,” The American Statistician, Vol. 44, No. 3, 1990,
pp. 245-249. doi:10.2307/2685351
()
() ()
()
()
()
()( )
1, 211,
,
,1
21,1,
,
01
z
z
Bnm zInm
Bnm
Bnm zI nm
Bnm
z
−−
+−
<<
[4] A. I. Zayed, “Handbook of Function and Generalized
Function Transformations,” CRC Press, London, 1996.
[5] A. R. Soltani and H. Homei, “A Generalization for Two-
Sided Power Distributions and Adjusted Method of Mo-
ments,” Statistics, Vol. 43, No. 6, 2009, pp. 611-620.
doi:10.1080/02331880802689506
where
(
,
z
)
I
ab is incomplete Beta function: